Bounding graphical t-wise balanced designs

Let X be the edges of the complete graph Kn on n vertices, provided with the natural action of Sn, the automorphism group of Kn. A t-wise balanced design (X, B) with parameters t-((2n), K, λ) is said to be graphical if B is fixed under the action of Sn. We show that for any pair (t, λ) with t > 1 or λ odd, there cannot exist a non-trivial graphical t-((2n), K, λ) design with n ⩾ 2t + λ + 4. Thus, in particular, for each such pair (t, λ) there are only a finite number of non-trivial graphical t-(v, K, λ) designs. If we further assume no repeated blocks, then for all cases with t > 1 or λ ≠ 2, there do not exist non-trivial graphical t-((2n), K, λ) designs with n ⩾ 2t + λ + 4.